Molecular transport across fluid interfaces: Coupling between solute dynamics and interface fluctuations

We investigate the transport mechanism of a small hydrophobic solute molecule across two types of fluid interfaces, (i) an interface between two immiscible liquids and (ii) a surfactant-covered liquid-liquid interface. These systems are modeled by coarse-grained molecular dynamics simulations. It is demonstrated that the dynamics of the solute molecule near the interface significantly deviates from Markovian Brownian motion. Specifically, the correlation time of the random force acting on the solute strongly depends on the distance between the solute and the interface and increases by two orders of magnitude within a very narrow (less than $1\mathrm{nm}$ wide) region near the interface. The slow fluctuations of the random force in this narrow region are caused by capillary waves. The region location and width are determined by interface protrusions caused by attraction between the solute and the hydrophobic phase. We use results of molecular dynamics simulations to develop a stochastic model for the coupled solute-interface dynamics and estimate the rate of the solute transport across the interface. The observed phenomenon appears to be a general feature of mass transport across fluid or flexible membranes. The coupling between the solute transport and the interface fluctuations is the strongest in areas corresponding to a large free energy gradient or near a free energy barrier for the solute transport. This suggests a strong influence of the coupled solute-interface dynamics on the rate of mass transfer across interfaces.

Figure 1

Density profiles of (a) surfactant-free hexadecane-water interface and (b) hexadecane-water interface covered by ${\mathrm{H}}_3{\mathrm{T}}_3$ surfactants [26].

Figure 2

ACF of the random force $\Gamma (t;z_{\mathrm{s}})$ acting on the solute near the surfactant-free oil-water interface. In this plot, ACF are normalized so that $C(t=0;z_{\mathrm{s}})=1$. The solute is constrained at (a) $z_{\mathrm{s}}=-1.06\mathrm{nm}$ (solid line), (b) $z_{\mathrm{s}}=0.59\mathrm{nm}$ (dashed line), and (c) $z_{\mathrm{s}}=1.75\mathrm{nm}$ (dashed-dotted line). The inset shows $C(t;z_{\mathrm{s}})$ for small $t$.

Figure 3

(a) Correlation time ${\tau }_f\left(z_{\mathrm{s}}\right)$ of the slowest fluctuations of the random force $\Gamma (z_{\mathrm{s}};t)$ acting on the solute constrained at $z=z_{\mathrm{s}}$ near surfactant-free and surfactant-covered interfaces; (b) corresponding free energy profiles.

Figure 4

Average dividing surfaces $z=h^{\left(0\right)}(r;z_{\mathrm{s}})$ corresponding to the solute constrained at different $z=z_{\mathrm{s}}$ near the (a) surfactant-free oil-water interface and (b) ${\mathrm{H}}_3{\mathrm{T}}_3$ monolayer. To emphasize rotational symmetry of the dividing surfaces, symmetric reflections of these surfaces with respect to the $z$ axis are also plotted. Dividing surfaces exhibiting significant deviations from the unperturbed planar interface are shown by dashed lines and the corresponding solute locations are shown by hollow circles. Solute locations which do not lead to significant interface perturbations are indicated by filled circles and the corresponding dividing surfaces are shown by solid lines. The gray (black) lines and circles correspond to solute located on the oil- (water-) rich sides of the interface.

Figure 5

Examples of ACF $C_{\mathbf{k}}(\tau ;z_{\mathrm{s}})$ of the interface modes for the surfactant-free oil-water interface $(z_{\mathrm{s}}=0.59\mathrm{nm})$ and ${\mathrm{H}}_3{\mathrm{T}}_3$ monolayer $(z_{\mathrm{s}}=0.67\mathrm{nm})$. Thick gray lines show exponential fits to $C_{\mathbf{k}}(\tau ;z_{\mathrm{s}})$. In this plot, ACF are normalized so that $C_{\mathbf{k}}(\tau =0;z_{\mathrm{s}})=1$.

Figure 6

Dependence of (a) ${\widehat{h}}_{\mathbf{k}}^{\left(0\right)}$, (b) ${\alpha }_{\mathbf{k}}$, and (c) ${\tau }_{\mathbf{k}}$ on $z_{\mathrm{s}}$ for a wave vector $\mathbf{k}$ with magnitude $0.64{\mathrm{nm}}^{-1}$.

Figure 7

Solute friction coefficients ${\gamma }_{\mathrm{MD}}$ and ${\gamma }_{\mathrm{s}}$ obtained from the ACF of forces ${\Gamma }_{\mathrm{MD}}(t;z_{\mathrm{s}})$ and ${\Gamma }_{\mathrm{s}}(t;z_{\mathrm{s}})$ are shown by thick gray and thin black lines, respectively. The friction coefficients at the surfactant-free oil-water interface and ${\mathrm{H}}_3{\mathrm{T}}_3$ monolayer are shown by dashed and solid lines, respectively.

Figure 8

Projection of the free energy $G(z_{\mathrm{s}},\left\{{\widehat{h}}_{\mathbf{k}}\right\})$ on the plane $z_{\mathrm{s}}-{\widehat{h}}_{\mathbf{k}}$ with $\mid k\mid =0.64{\mathrm{nm}}^{-1}$ for (a) surfactant-free oil-water interface and (b) surfactant monolayer. Minimal energy paths are shown by thick solid lines and the paths assumed in the calculation of the mean transport time $T[{\gamma }_{\mathrm{s}},\tilde{G}]$ are shown by dashed lines. Points $z_{h,\max }$ corresponding to maxima of ${\widehat{h}}_{\mathbf{k}}^{\left(0\right)}\left(z_{\mathrm{s}}\right)$ are shown by circles and points $z_{G,\max }$ corresponding to maxima of $G_0\left(z_{\mathrm{s}}\right)$ are shown by squares.